Semi-nonoscillation intervals in the analysis of sign constancy of Green’s functions of Dirichlet, Neumann and focal impulsive problems
- Alexander Domoshnitsky^{1}Email author and
- Guy Landsman^{2}
https://doi.org/10.1186/s13662-017-1134-1
© The Author(s) 2017
Received: 1 December 2016
Accepted: 9 March 2017
Published: 20 March 2017
Abstract
Keywords
impulsive equations Green’s functions positivity/negativity of Green’s functions boundary value problem second orderMSC
34K10 34B37 34A40 34A37 34K481 Introduction
Impulsive equations attract attention of many recognized mathematicians. See, for example, the books [1–6]. The positivity of solutions to the Dirichlet problem was studied in [7]. A generalized Dirichlet problem was considered in [7–9]. Multipoint problems and problems with integral boundary conditions were considered in [10–15]. The Dirichlet problem for impulsive equations with impulses at variable moments was studied in [16]. All these works considered impulsive ordinary differential equations.
Let us assume that all trajectories of solutions to a non-impulsive ordinary differential equation are known. In this case, impulses imply only choosing the trajectory between the points of impulses, but we stay on the trajectory of a corresponding solution of the non-impulsive equation between \(t_{i}\) and \(t_{i+1}\). In the case of an impulsive equation with delay, it is not true anymore. That is why the properties of delay impulsive equations can be quite different.
There are only a few results on the positivity of solutions to impulsive differential equations with delay. Note the results [17–19] about the positivity of Green’s functions for boundary value problems for first order delay impulsive equations. Nonoscillation of second order delay impulsive differential equation was studied in [20]. Sturmian comparison theory for impulsive second order delay equations was studied in [21]. The positivity of Green’s functions for the nth order impulsive delay differential equation was considered in [22]. The idea of construction of Green’s functions for second order impulsive differential equations was first proposed in [23]. The use of Green’s functions for auxiliary impulsive problems in the study of sign constancy of delay impulsive differential equations was proposed in [24], where a one-point problem was studied. Note also paper [25] for focal problems.
In this paper we use the results for one-point and focal problems in order to obtain results on the sign constancy for Green’s functions of two-point boundary value problems.
Let D be a space of functions \(x:[0,\omega]\rightarrow{\mathbb {R}}\) such that their derivative \(x'(t)\) is absolutely continuous on every interval \(t\in{[t_{i},t_{i+1})}\), \(i=0,\ldots,r, x''\in {L_{\infty}}\), there exist the finite limits \(x(t_{i}-0)= \lim_{t\rightarrow{t_{i}^{-}}} x(t)\) and \(x'(t_{i}-0)= \lim_{t\rightarrow {t_{i}^{-}}} x'(t)\) and condition (1.2) is satisfied at points \(t_{i}\) (\(i=0,\ldots,r\)). We understand solution x as a function \(x\in{D}\) satisfying (1.1)-(1.3).
Definition 1.1
We call \([0,\omega]\) a semi-nonoscillation interval of \((Lx)(t)=0\) if every nontrivial solution having zero of derivative does not have zero on this interval.
The influence of nonoscillation on sign properties of Green’s functions in the case of nth order differential equations was found in the known papers [26, 27]. An extension of these results on delay differential equations was obtained in [28, 29]. The importance of a semi-nonoscillation interval in the case of non-impulsive delay differential equations was first noted in [30]. In this paper we develop the use of semi-nonoscillation intervals to impulsive delay differential equations.
2 Construction of Green’s functions
We denote by \(G_{i}(t,s)\) Green’s function of problem (1.1)-(1.3), (2.i) respectively.
Lemma 2.1
- (1)
\(b_{j}(t)\leq{0}, t\in{[0,\omega]}\);
- (2)the Cauchy function \(C_{1}(t,s)\) of the first order equationis positive for \(0\leq{s}\leq{t}\leq{\omega}\).$$ \textstyle\begin{cases} y'(t)+\sum_{j=1}^{p}{a_{j}(t)y(t-\tau_{j}(t))}=0,\quad t\in{[0,\omega]}, \\ y(t_{k})=\delta_{k} y(t_{k}-0),\quad k=1,\ldots,m, \\ y(\zeta)=0,\quad \zeta< 0, \end{cases} $$(2.16)
Then the Cauchy function \(C(t,s)\) of equation (1.1) and its derivative \(C'_{t}(t,s)\) are positive in \(0\leq{s}\leq{t}\leq {\omega}\).
Proof
Lemma 2.2
If the conditions (1) and (2) of Lemma 2.1 are fulfilled, then Green’s function \(G_{2}(t,s)\) of (1.1)-(1.3), (2.2) exists and there exists an interval \((0,\epsilon_{s})\) such that \(G_{2}(t,s)<0\) for \(t\in(0,\epsilon_{s})\).
Proof
According to Lemma 2.1, we have \(C'_{t}(t,s)>0\) in \(0\leq{s}\leq{t}\leq{\omega}\). This implies that Green’s function \(G_{2}(t,s)\) of (2.15), (1.2), (1.3), (2.2), which is defined by (2.12), exists.
Lemma 2.2 has been proven. □
Lemma 2.3
If the conditions (1) and (2) of Lemma 2.1 are fulfilled, then Green’s function \(G_{3}(t,s)\) of (1.1)-(1.3), (2.3) exists and there exists an interval \((0,\epsilon_{s})\) such that \(G_{3}(t,s)<0\) for \(t\in(0,\epsilon_{s})\).
Proof
Lemma 2.4
If the conditions (1) and (2) of Lemma 2.1 are fulfilled, then Green’s function \(G_{4}(t,s)\) of (1.1)-(1.3), (2.4) exists and there exists an interval \((0,\epsilon_{s})\) such that \(G_{4}(t,s)<0\) for \(t\in(0,\epsilon_{s})\).
Proof
Let us demonstrate that the problem \((Lx)(t)=0, x(0)=0, x(\omega)=0\) has only the trivial solution. If there exists a nontrivial solution of this problem, it is proportional to \(C(t,0)\). According to Lemma 2.1, \(C(t,0)>0\) for \(t\in{(0,\omega]}\). It means that \(x(\omega)=C(\omega,0)>0\). That contradicts the assumption \(x(\omega)=0\).
Lemma 2.4 has been proven. □
3 Sign constancy of Green’s functions
In this section we will prove the sign constancy of Green’s functions \(G_{4}(t,s)\) and \(G_{5}(t,s)\) using the results from [24] and [25] about the sign constancy of \(G_{1}(t,s), G_{2}(t,s)\) and \(G_{3}(t,s)\).
Theorem 3.1
- (1)
\(G_{1}^{\xi}(t,s)\geq{0}, t,s\in{[0,\xi]}\) for every \(0<\xi<\omega\).
- (2)
\([0,\omega]\) is a semi-nonoscillation interval of \((Lx)(t)=0\).
- (3)
\(b_{j}(t)\leq{0}, t\in{[0,\omega]}\).
- (4)
The Cauchy function \(C_{1}(t,s)\) of the first order equation (2.16) is positive for \(0\leq{s}\leq{t}\leq{\omega}\).
Proof
Step 2. Let us assume that there exists a solution \(x(t)\) changing sign on \([0,\omega]\) for nonnegative \(f(t)\). We have to consider two cases: the solution \(x(t)\) changes sign first time from positive to negative; and the solution \(x(t)\) changes sign first from negative to positive.
In the first case, we have a point η such that \(x(\eta)=0\). It means that our function \(x(t)\) satisfies the problem \((Lx)(t)=f(t), x'(0)=0, x(\eta)=0\). We have proven above that \(G_{3}(t,s)\leq{0}\) and this excludes the possibility of \(x(t)>0\) for \(t\in{[0,\eta)}\).
Theorem 3.1 has been proven. □
Theorem 3.2
Proof
It is clear that all the conditions of assertion (1) of Theorem 4.1 from [24] are fulfilled. According to this theorem, \(G_{1}^{\xi}(t,s)\geq{0}\) for every \(t,s\in(0,\omega)\) and every \(0<\xi<\omega\). Using Theorem 3.1 above, we obtain that \(G_{2}(t,s)\leq{0}\), \(G_{3}(t,s)\leq{0}\), \(G_{4}(t,s)\leq{0}\) for \(t,s\in {[0,\omega]}\). If, in addition, \(\sum_{j=1}^{p}{b_{j}(t)\chi(t-\theta _{j}(t))}\not\equiv{0}, t\in{[0,\omega]}\), then it follows that \(G_{5}(t,s)\leq{0}\) for \(t,s\in{[0,\omega]}\).
Theorem 3.2 has been proven. □
Theorem 3.3
- (1)
\(G_{2}^{\xi}(t,s)\leq{0}, t,s\in{[0,\xi]}\) for every \(0<\xi<\omega\).
- (2)
\([0,\omega]\) is a semi-nonoscillation interval of \((Lx)(t)=0\).
Proof
Let us consider problem (1.1)-(1.3), (2.4). According to Lemma 2.4, there exists a unique solution for every summable \(f(t)\). Let us assume that \(G_{4}(t,s)\) changes sign. It means that there exists a function \(f(t)\geq{0}\) such that the solution \(x(t)\) changes sign from negative to positive according to Lemma 2.4. Then there is a point \(0<\xi<\omega\) such that \(x(\xi)=\alpha>0\) and \(x'(\xi)=0\) (see Figure 1). From condition 1, we know that Green’s function for this problem, \(G_{2}^{\xi}(t,s)\) is nonpositive. From condition 2, it follows that the solution of problem \((Lx)(t)=0, x(\xi )=\alpha>0, x'(\xi)=0\) is positive for \(t\in{(0,\xi]}\). Then \(x(t)\leq{0}\) for \(t\in{[0,\xi]}\). This contradicts Lemma 2.4, which claims that \(x(t)\) can change its sign only from negative to positive for nonnegative \(f(t)\). Then \(G_{4}(t,s)\) should be nonpositive.
Theorem 3.3 has been proven. □
Theorem 3.4
Proof
Looking at Theorem 5.1 from [25], we can see that the problem satisfies all of the conditions. Then \(G_{2}(t,s)\leq{0}\) for every \(t,s\in(0,\omega)\). Using Theorem 3.3 above, we obtain that \(G_{4}(t,s)\leq{0}\) for \(t,s\in{[0,\omega]}\).
Theorem 3.4 has been proven. □
Example 3.5
Theorem 3.6
Proof
Theorem 3.6 has been proven. □
In the particular case \(a_{j}(t)=0, j=1,\ldots,p\), we have
Corollary 3.7
Theorem 3.8
Proof
Theorem 3.8 has been proven. □
Theorem 3.9
Example 3.10
Corollary 3.11
Proof
Declarations
Acknowledgements
Dr. Shlomo Yanetz (Bar Ilan University, Israel) for his important and valuable remarks. This paper is a part of the second author’s Ph.D. thesis which is being carried out in the Department of Mathematics at Bar-Ilan University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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